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Properties

Label 2-845-65.47-c1-0-52
Degree $2$
Conductor $845$
Sign $0.374 + 0.927i$
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 + (1 − i)3-s + 4-s + (2 + i)5-s + (−1 − i)6-s + 2·7-s − 3i·8-s + i·9-s + (1 − 2i)10-s + (1 − i)11-s + (1 − i)12-s − 2i·14-s + (3 − i)15-s − 16-s + (−1 + i)17-s + 18-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.577 − 0.577i)3-s + 0.5·4-s + (0.894 + 0.447i)5-s + (−0.408 − 0.408i)6-s + 0.755·7-s − 1.06i·8-s + 0.333i·9-s + (0.316 − 0.632i)10-s + (0.301 − 0.301i)11-s + (0.288 − 0.288i)12-s − 0.534i·14-s + (0.774 − 0.258i)15-s − 0.250·16-s + (−0.242 + 0.242i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27030 - 1.53141i\)
\(L(\frac12)\) \(\approx\) \(2.27030 - 1.53141i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 + (5 - 5i)T - 19iT^{2} \)
23 \( 1 + (3 + 3i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + (-7 - 7i)T + 41iT^{2} \)
43 \( 1 + (-1 - i)T + 43iT^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (7 + 7i)T + 59iT^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + (1 + i)T + 71iT^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-5 - 5i)T + 89iT^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−10.26366995784878187772740868213, −9.316605711841791883502725019831, −8.254437816824936412724093803855, −7.59713622222743386894181907785, −6.51622421284138473719184417440, −5.90801490141797863049618257812, −4.41874111017273617537952364698, −3.16149829751021263068626567708, −2.08085533926447441370692004528, −1.66225632278128860196429221082, 1.71595409120037686296328766557, 2.72530759282490643504847108959, 4.24164419036161304618173546648, 5.09615920577456877081399451180, 6.05853080805161693354660420583, 6.85561182602446446940695513032, 7.82473840026792697154831085425, 8.893044183789676757927861983365, 9.102382348884222298631832276886, 10.29787544650982573590038777625

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