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Properties

Label 2-845-65.28-c1-0-64
Degree $2$
Conductor $845$
Sign $-0.907 + 0.419i$
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  + (2.29 − 1.32i)2-s + (0.335 − 1.25i)3-s + (2.51 − 4.34i)4-s + (−1.81 − 1.30i)5-s + (−0.889 − 3.31i)6-s + (−0.0561 + 0.0972i)7-s − 8.00i·8-s + (1.14 + 0.658i)9-s + (−5.89 − 0.585i)10-s + (−0.479 + 1.78i)11-s + (−4.60 − 4.60i)12-s + 0.297i·14-s + (−2.24 + 1.83i)15-s + (−5.58 − 9.67i)16-s + (−2.63 + 0.706i)17-s + 3.49·18-s + ⋯
L(s)  = 1  + (1.62 − 0.936i)2-s + (0.193 − 0.723i)3-s + (1.25 − 2.17i)4-s + (−0.812 − 0.583i)5-s + (−0.363 − 1.35i)6-s + (−0.0212 + 0.0367i)7-s − 2.83i·8-s + (0.380 + 0.219i)9-s + (−1.86 − 0.185i)10-s + (−0.144 + 0.539i)11-s + (−1.32 − 1.32i)12-s + 0.0795i·14-s + (−0.579 + 0.474i)15-s + (−1.39 − 2.41i)16-s + (−0.639 + 0.171i)17-s + 0.823·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.907 + 0.419i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.907 + 0.419i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794430 - 3.60963i\)
\(L(\frac12)\) \(\approx\) \(0.794430 - 3.60963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.81 + 1.30i)T \)
13 \( 1 \)
good2 \( 1 + (-2.29 + 1.32i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.335 + 1.25i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.0561 - 0.0972i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.479 - 1.78i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.63 - 0.706i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.72 + 1.80i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.10 + 0.831i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.03 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.624 + 0.624i)T - 31iT^{2} \)
37 \( 1 + (-0.737 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.24 - 1.40i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.00 - 3.76i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 0.345T + 47T^{2} \)
53 \( 1 + (3.59 + 3.59i)T + 53iT^{2} \)
59 \( 1 + (0.332 + 1.24i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.39 + 2.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.124 - 0.0721i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.41 - 5.28i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 9.06iT - 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (0.549 + 0.147i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.9 - 7.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.12820580649799583915795286112, −9.194644636926210437933219424319, −7.79558309902492496714691104451, −7.16001674266519600396570003922, −6.09975358172672587590107080270, −4.98300079295462196042778397050, −4.41066552704310677017729342400, −3.38511425743554146208348108701, −2.24648481871227322633811679537, −1.15999794376129867925438911345, 2.75673271966619448258117154613, 3.68760713752667982342423155090, 4.12355613110602929984104479015, 5.18439846547061191776262137297, 6.05734580128983081210166908273, 7.07669070613934112667835122311, 7.58787695922580626936114688591, 8.544244022442877807402479046263, 9.734512533712793469608257774629, 10.85576465196533939675682965450

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