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Properties

Label 1-293-293.82-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.815 - 0.579i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.397 + 0.917i)2-s + (−0.976 + 0.213i)3-s + (−0.683 − 0.729i)4-s + (−0.999 − 0.0430i)5-s + (0.192 − 0.981i)6-s + (0.772 + 0.635i)7-s + (0.941 − 0.337i)8-s + (0.908 − 0.417i)9-s + (0.436 − 0.899i)10-s + (0.107 + 0.994i)11-s + (0.823 + 0.566i)12-s + (−0.925 − 0.377i)13-s + (−0.890 + 0.455i)14-s + (0.985 − 0.171i)15-s + (−0.0645 + 0.997i)16-s + (−0.397 + 0.917i)17-s + ⋯
L(s)  = 1  + (−0.397 + 0.917i)2-s + (−0.976 + 0.213i)3-s + (−0.683 − 0.729i)4-s + (−0.999 − 0.0430i)5-s + (0.192 − 0.981i)6-s + (0.772 + 0.635i)7-s + (0.941 − 0.337i)8-s + (0.908 − 0.417i)9-s + (0.436 − 0.899i)10-s + (0.107 + 0.994i)11-s + (0.823 + 0.566i)12-s + (−0.925 − 0.377i)13-s + (−0.890 + 0.455i)14-s + (0.985 − 0.171i)15-s + (−0.0645 + 0.997i)16-s + (−0.397 + 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.815 - 0.579i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.815 - 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07602204360 + 0.2381422658i\)
\(L(\frac12)\) \(\approx\) \(-0.07602204360 + 0.2381422658i\)
\(L(1)\) \(\approx\) \(0.3658041885 + 0.2840048944i\)
\(L(1)\) \(\approx\) \(0.3658041885 + 0.2840048944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.397 + 0.917i)T \)
3 \( 1 + (-0.976 + 0.213i)T \)
5 \( 1 + (-0.999 - 0.0430i)T \)
7 \( 1 + (0.772 + 0.635i)T \)
11 \( 1 + (0.107 + 0.994i)T \)
13 \( 1 + (-0.925 - 0.377i)T \)
17 \( 1 + (-0.397 + 0.917i)T \)
19 \( 1 + (0.985 - 0.171i)T \)
23 \( 1 + (0.192 + 0.981i)T \)
29 \( 1 + (-0.618 - 0.785i)T \)
31 \( 1 + (-0.999 + 0.0430i)T \)
37 \( 1 + (-0.991 - 0.128i)T \)
41 \( 1 + (-0.474 - 0.880i)T \)
43 \( 1 + (0.512 - 0.858i)T \)
47 \( 1 + (-0.954 + 0.296i)T \)
53 \( 1 + (0.107 + 0.994i)T \)
59 \( 1 + (0.357 - 0.933i)T \)
61 \( 1 + (-0.847 - 0.530i)T \)
67 \( 1 + (-0.744 + 0.668i)T \)
71 \( 1 + (-0.317 + 0.948i)T \)
73 \( 1 + (-0.683 + 0.729i)T \)
79 \( 1 + (-0.847 - 0.530i)T \)
83 \( 1 + (-0.798 + 0.601i)T \)
89 \( 1 + (-0.847 + 0.530i)T \)
97 \( 1 + (0.584 - 0.811i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.44440383899855766353871479421, −24.06027668186918299641624054621, −22.833646812405754957802392963944, −22.331695518172537564058659720003, −21.28731222896494640608008587437, −20.28496023194426095809307106025, −19.46188314594691085993802774430, −18.49454011086812845628607761660, −17.90353597823664385619198214401, −16.54977452809452427950934831167, −16.42542127760800202576470364891, −14.61515783061787435913981786312, −13.55051134083426701625812265083, −12.419775152193782365490822945819, −11.52561795341116716958071916021, −11.19103658463187191625511280731, −10.194885162174609025706434459978, −8.851216397475564870043565294251, −7.67302593523726628675348776077, −7.04168884327558398597971708610, −5.11573749429926652533350245511, −4.40678452557107531980529906971, −3.16863231446896703030002123806, −1.469562233362189922399144913858, −0.23600096511727005434175224001, 1.60712739666255153039284938031, 3.991728149787218951238484839627, 4.95681506859741443348091907881, 5.6332267164558537199077449903, 7.13448341036425432643898634724, 7.608679006216400009029307495769, 8.90851395236074292368759582211, 9.94457378430760565131328797686, 11.036436535095309790533572535964, 11.964061409307044153747716152816, 12.834372169099270903192637197013, 14.52208817165823107029118325466, 15.388998734409744577684579028350, 15.67012368404285340192835684337, 17.08105710379708665826975574808, 17.52683901069041306989041914586, 18.4475639114124378431551396272, 19.39870813553330664020006105955, 20.477365239136800448287802419452, 21.9664105253703815681899962051, 22.53325980705579399363633183310, 23.49461263138213572595012027572, 24.16739604396107599309315089913, 24.81490058422862437893974080079, 26.14223435712027280154072473713

Graph of the $Z$-function along the critical line