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Properties

Label 2-845-65.4-c1-0-33
Degree $2$
Conductor $845$
Sign $0.0640 - 0.997i$
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  + (−0.165 + 0.286i)2-s + (2.33 + 1.34i)3-s + (0.945 + 1.63i)4-s + (0.702 − 2.12i)5-s + (−0.771 + 0.445i)6-s + (1.67 + 2.90i)7-s − 1.28·8-s + (2.12 + 3.67i)9-s + (0.492 + 0.552i)10-s + (−2.81 − 1.62i)11-s + 5.08i·12-s − 1.10·14-s + (4.49 − 4.00i)15-s + (−1.67 + 2.90i)16-s + (−1.68 + 0.974i)17-s − 1.40·18-s + ⋯
L(s)  = 1  + (−0.116 + 0.202i)2-s + (1.34 + 0.777i)3-s + (0.472 + 0.818i)4-s + (0.314 − 0.949i)5-s + (−0.314 + 0.181i)6-s + (0.633 + 1.09i)7-s − 0.455·8-s + (0.707 + 1.22i)9-s + (0.155 + 0.174i)10-s + (−0.847 − 0.489i)11-s + 1.46i·12-s − 0.296·14-s + (1.16 − 1.03i)15-s + (−0.419 + 0.726i)16-s + (−0.409 + 0.236i)17-s − 0.331·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0640 - 0.997i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.0640 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94719 + 1.82619i\)
\(L(\frac12)\) \(\approx\) \(1.94719 + 1.82619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.15279317782104292587205097600, −9.120505088410157887919248282962, −8.645689639787038290323981538097, −8.270004875520543760617556233038, −7.37537525912442968049500839811, −5.85952330994325976161392371188, −4.98239081709930535470418900806, −3.93056714165324405479625016730, −2.81207602325096388143082407726, −2.10908524023404822883622404228, 1.27760421043448916365913303611, 2.33304827713636053034469613552, 2.99461876226113567240538373431, 4.43115697434380663507297828650, 5.79657167962378745221338375556, 7.01026949264433511525088574197, 7.27058352719880955737834768068, 8.113980190612072653130303642273, 9.259866184688509464962113976045, 10.03266402718838103023884306447

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