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Properties

Label 2-845-65.57-c1-0-47
Degree $2$
Conductor $845$
Sign $-0.347 + 0.937i$
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  − 2.08·2-s + (1.94 − 1.94i)3-s + 2.35·4-s + (0.194 + 2.22i)5-s + (−4.06 + 4.06i)6-s − 2.91i·7-s − 0.750·8-s − 4.59i·9-s + (−0.405 − 4.65i)10-s + (−0.0186 − 0.0186i)11-s + (4.59 − 4.59i)12-s + 6.07i·14-s + (4.71 + 3.96i)15-s − 3.15·16-s + (2.02 − 2.02i)17-s + 9.58i·18-s + ⋯
L(s)  = 1  − 1.47·2-s + (1.12 − 1.12i)3-s + 1.17·4-s + (0.0869 + 0.996i)5-s + (−1.66 + 1.66i)6-s − 1.10i·7-s − 0.265·8-s − 1.53i·9-s + (−0.128 − 1.47i)10-s + (−0.00561 − 0.00561i)11-s + (1.32 − 1.32i)12-s + 1.62i·14-s + (1.21 + 1.02i)15-s − 0.787·16-s + (0.491 − 0.491i)17-s + 2.26i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.347 + 0.937i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.347 + 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561547 - 0.807191i\)
\(L(\frac12)\) \(\approx\) \(0.561547 - 0.807191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.194 - 2.22i)T \)
13 \( 1 \)
good2 \( 1 + 2.08T + 2T^{2} \)
3 \( 1 + (-1.94 + 1.94i)T - 3iT^{2} \)
7 \( 1 + 2.91iT - 7T^{2} \)
11 \( 1 + (0.0186 + 0.0186i)T + 11iT^{2} \)
17 \( 1 + (-2.02 + 2.02i)T - 17iT^{2} \)
19 \( 1 + (3.38 + 3.38i)T + 19iT^{2} \)
23 \( 1 + (-0.262 - 0.262i)T + 23iT^{2} \)
29 \( 1 + 4.18iT - 29T^{2} \)
31 \( 1 + (-0.835 + 0.835i)T - 31iT^{2} \)
37 \( 1 + 6.45iT - 37T^{2} \)
41 \( 1 + (-5.54 + 5.54i)T - 41iT^{2} \)
43 \( 1 + (-4.90 - 4.90i)T + 43iT^{2} \)
47 \( 1 + 0.833iT - 47T^{2} \)
53 \( 1 + (-0.902 + 0.902i)T - 53iT^{2} \)
59 \( 1 + (-1.05 + 1.05i)T - 59iT^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + (2.61 - 2.61i)T - 71iT^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 + 4.25iT - 79T^{2} \)
83 \( 1 - 1.31iT - 83T^{2} \)
89 \( 1 + (2.36 - 2.36i)T - 89iT^{2} \)
97 \( 1 - 0.405T + 97T^{2} \)
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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

−9.710940620676113086734124143594, −9.057534954839039325586827396822, −8.097397033263100658065638783016, −7.42386948801720230552376116366, −7.15197147195936116082973928611, −6.23528092967888724967136170900, −4.14353272140436328659753656543, −2.86123845094540467860328289453, −2.00791248690236450103953666889, −0.69820598959118573283253288104, 1.59494548523134089463572449108, 2.70809932383894152562992206115, 4.05259910699268296037885141330, 5.00708370678276352325117252985, 6.15475741512373805160669923372, 7.77448111371819871209686278584, 8.305521963888545735903426866094, 8.946967064178231276985985442737, 9.286775024324742328183244811939, 10.10787400097140037694891308392

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