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Properties

Label 2-845-65.57-c1-0-32
Degree $2$
Conductor $845$
Sign $0.956 + 0.292i$
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  − 0.274·2-s + (1.67 − 1.67i)3-s − 1.92·4-s + (−1.69 + 1.45i)5-s + (−0.459 + 0.459i)6-s − 0.386i·7-s + 1.07·8-s − 2.58i·9-s + (0.466 − 0.399i)10-s + (3.08 + 3.08i)11-s + (−3.21 + 3.21i)12-s + 0.106i·14-s + (−0.409 + 5.26i)15-s + 3.55·16-s + (1.39 − 1.39i)17-s + 0.710i·18-s + ⋯
L(s)  = 1  − 0.194·2-s + (0.964 − 0.964i)3-s − 0.962·4-s + (−0.759 + 0.650i)5-s + (−0.187 + 0.187i)6-s − 0.145i·7-s + 0.381·8-s − 0.861i·9-s + (0.147 − 0.126i)10-s + (0.929 + 0.929i)11-s + (−0.928 + 0.928i)12-s + 0.0283i·14-s + (−0.105 + 1.36i)15-s + 0.888·16-s + (0.338 − 0.338i)17-s + 0.167i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.956 + 0.292i$
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: \((2,\ 845,\ (\ :1/2),\ 0.956 + 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45410 - 0.217070i\)
\(L(\frac12)\) \(\approx\) \(1.45410 - 0.217070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.69 - 1.45i)T \)
13 \( 1 \)
good2 \( 1 + 0.274T + 2T^{2} \)
3 \( 1 + (-1.67 + 1.67i)T - 3iT^{2} \)
7 \( 1 + 0.386iT - 7T^{2} \)
11 \( 1 + (-3.08 - 3.08i)T + 11iT^{2} \)
17 \( 1 + (-1.39 + 1.39i)T - 17iT^{2} \)
19 \( 1 + (-3.54 - 3.54i)T + 19iT^{2} \)
23 \( 1 + (-0.235 - 0.235i)T + 23iT^{2} \)
29 \( 1 + 8.16iT - 29T^{2} \)
31 \( 1 + (-2.54 + 2.54i)T - 31iT^{2} \)
37 \( 1 + 4.82iT - 37T^{2} \)
41 \( 1 + (-3.29 + 3.29i)T - 41iT^{2} \)
43 \( 1 + (-4.82 - 4.82i)T + 43iT^{2} \)
47 \( 1 - 9.83iT - 47T^{2} \)
53 \( 1 + (7.17 - 7.17i)T - 53iT^{2} \)
59 \( 1 + (1.71 - 1.71i)T - 59iT^{2} \)
61 \( 1 - 10.6T + 61T^{2} \)
67 \( 1 - 6.37T + 67T^{2} \)
71 \( 1 + (-3.07 + 3.07i)T - 71iT^{2} \)
73 \( 1 + 6.08T + 73T^{2} \)
79 \( 1 + 3.34iT - 79T^{2} \)
83 \( 1 - 5.18iT - 83T^{2} \)
89 \( 1 + (3.53 - 3.53i)T - 89iT^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−9.787922784575861264334798247619, −9.330588089450223727856998109231, −8.228184004768337289504584661976, −7.67896814998026013118957576650, −7.15960796209637546038666708900, −5.97482226975451609167223648678, −4.43171290864223558200988270536, −3.72429261762326339688647808550, −2.54283416682228589802746198545, −1.09763228983960632123458223272, 0.983849335851987929947415470401, 3.22514537698705178617135003134, 3.76973006331369210356460287769, 4.66095669673330954286143671338, 5.45863760265636806385286109697, 7.03314909041721656464623799571, 8.303437850597873313894324325732, 8.585345585956393771950841916760, 9.230052587028783663129543948705, 9.872651824787661060485748022588

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