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Properties

Label 2-845-65.29-c1-0-49
Degree $2$
Conductor $845$
Sign $0.976 + 0.215i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.286 + 0.165i)2-s + (2.33 − 1.34i)3-s + (−0.945 + 1.63i)4-s + (2.12 − 0.702i)5-s + (−0.445 + 0.771i)6-s + (2.90 + 1.67i)7-s − 1.28i·8-s + (2.12 − 3.67i)9-s + (−0.492 + 0.552i)10-s + (−1.62 − 2.81i)11-s + 5.08i·12-s − 1.10·14-s + (4.00 − 4.49i)15-s + (−1.67 − 2.90i)16-s + (1.68 + 0.974i)17-s + 1.40i·18-s + ⋯
L(s)  = 1  + (−0.202 + 0.116i)2-s + (1.34 − 0.777i)3-s + (−0.472 + 0.818i)4-s + (0.949 − 0.314i)5-s + (−0.181 + 0.314i)6-s + (1.09 + 0.633i)7-s − 0.455i·8-s + (0.707 − 1.22i)9-s + (−0.155 + 0.174i)10-s + (−0.489 − 0.847i)11-s + 1.46i·12-s − 0.296·14-s + (1.03 − 1.16i)15-s + (−0.419 − 0.726i)16-s + (0.409 + 0.236i)17-s + 0.331i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49960 - 0.272176i\)
\(L(\frac12)\) \(\approx\) \(2.49960 - 0.272176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.12 + 0.702i)T \)
13 \( 1 \)
good2 \( 1 + (0.286 - 0.165i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-2.33 + 1.34i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.90 - 1.67i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.62 + 2.81i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.68 - 0.974i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.622 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.33 - 1.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.78T + 31T^{2} \)
37 \( 1 + (1.68 - 0.974i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.39 - 2.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.56 + 4.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.86iT - 47T^{2} \)
53 \( 1 - 12.8iT - 53T^{2} \)
59 \( 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.47 + 2.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.62 + 4.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.46iT - 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 8.61iT - 83T^{2} \)
89 \( 1 + (5.15 + 8.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.56 + 2.63i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.730365355777596460887777918983, −8.978663130950022804064779375235, −8.401694247722648714096856757593, −7.994027473871426191380058947069, −7.07313461277039182173074926873, −5.79149562663453569639859476012, −4.81674679839676207890975290309, −3.40051463147182783865232635934, −2.52239693261705768346006057106, −1.46316482427040217296357018501, 1.61954020489815350220496790801, 2.45308088838035726474008260244, 3.89596817329624890932419067111, 4.80984332979022578329754140763, 5.51019520641258574370030450391, 6.95755307998823904349987450787, 8.007205168379979224539037237610, 8.649993403477274990688832267295, 9.615446165095119534290409278012, 10.05909149387738118063936183923

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