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Properties

Label 2-845-5.4-c1-0-21
Degree $2$
Conductor $845$
Sign $-0.894 - 0.447i$
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  + i·2-s + 2i·3-s + 4-s + (2 + i)5-s − 2·6-s + 3i·8-s − 9-s + (−1 + 2i)10-s − 2·11-s + 2i·12-s + (−2 + 4i)15-s − 16-s − i·18-s − 6·19-s + (2 + i)20-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.15i·3-s + 0.5·4-s + (0.894 + 0.447i)5-s − 0.816·6-s + 1.06i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s − 0.603·11-s + 0.577i·12-s + (−0.516 + 1.03i)15-s − 0.250·16-s − 0.235i·18-s − 1.37·19-s + (0.447 + 0.223i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475780 + 2.01543i\)
\(L(\frac12)\) \(\approx\) \(0.475780 + 2.01543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2 - i)T \)
13 \( 1 \)
good2 \( 1 - iT - 2T^{2} \)
3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−10.35134645824883003215451195087, −10.01910007183442033340494949047, −8.755498079935120604911876661826, −8.142116494652709980814741667129, −6.74978280820196409084840839384, −6.39615200058717896816035661040, −5.24770428263861635359632695637, −4.58291719282647597348243608599, −3.10073755491167276543285795263, −2.13076582645055884157808769733, 1.02764169643058473740593148586, 2.00485295887563018000026616432, 2.73947032183209890753205430055, 4.29337848099055378161232012987, 5.65560871970331382424842091566, 6.42642892346781123023563355893, 7.15744808508653996101297506864, 8.087711438496173037540703391661, 9.039565531077304240188068414120, 10.14369918590582487730351557528

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