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Properties

Label 2-845-65.57-c1-0-14
Degree $2$
Conductor $845$
Sign $-0.639 - 0.768i$
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  − 0.493·2-s + (−0.664 + 0.664i)3-s − 1.75·4-s + (2.21 − 0.284i)5-s + (0.328 − 0.328i)6-s + 3.67i·7-s + 1.85·8-s + 2.11i·9-s + (−1.09 + 0.140i)10-s + (0.486 + 0.486i)11-s + (1.16 − 1.16i)12-s − 1.81i·14-s + (−1.28 + 1.66i)15-s + 2.59·16-s + (1.67 − 1.67i)17-s − 1.04i·18-s + ⋯
L(s)  = 1  − 0.349·2-s + (−0.383 + 0.383i)3-s − 0.878·4-s + (0.991 − 0.127i)5-s + (0.134 − 0.134i)6-s + 1.38i·7-s + 0.655·8-s + 0.705i·9-s + (−0.346 + 0.0444i)10-s + (0.146 + 0.146i)11-s + (0.337 − 0.337i)12-s − 0.485i·14-s + (−0.331 + 0.429i)15-s + 0.648·16-s + (0.407 − 0.407i)17-s − 0.246i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.639 - 0.768i$
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: \((2,\ 845,\ (\ :1/2),\ -0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365949 + 0.780647i\)
\(L(\frac12)\) \(\approx\) \(0.365949 + 0.780647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.21 + 0.284i)T \)
13 \( 1 \)
good2 \( 1 + 0.493T + 2T^{2} \)
3 \( 1 + (0.664 - 0.664i)T - 3iT^{2} \)
7 \( 1 - 3.67iT - 7T^{2} \)
11 \( 1 + (-0.486 - 0.486i)T + 11iT^{2} \)
17 \( 1 + (-1.67 + 1.67i)T - 17iT^{2} \)
19 \( 1 + (3.87 + 3.87i)T + 19iT^{2} \)
23 \( 1 + (-0.957 - 0.957i)T + 23iT^{2} \)
29 \( 1 - 9.51iT - 29T^{2} \)
31 \( 1 + (4.81 - 4.81i)T - 31iT^{2} \)
37 \( 1 + 1.83iT - 37T^{2} \)
41 \( 1 + (0.391 - 0.391i)T - 41iT^{2} \)
43 \( 1 + (-1.53 - 1.53i)T + 43iT^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 + (2.47 - 2.47i)T - 53iT^{2} \)
59 \( 1 + (7.35 - 7.35i)T - 59iT^{2} \)
61 \( 1 - 6.19T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + (-4.74 + 4.74i)T - 71iT^{2} \)
73 \( 1 + 3.37T + 73T^{2} \)
79 \( 1 - 3.12iT - 79T^{2} \)
83 \( 1 - 2.13iT - 83T^{2} \)
89 \( 1 + (2.38 - 2.38i)T - 89iT^{2} \)
97 \( 1 + 7.07T + 97T^{2} \)
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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

−10.47141927776528523525823182392, −9.375473742107547606935465270783, −9.094784856680233834500215334158, −8.314468841072013258177027960688, −7.06546221540704612017770709708, −5.80014188110582605898251255844, −5.23747818797351430701249084005, −4.58064681008566145608735369738, −2.89707128416656376664543226567, −1.66456838500561549226747985037, 0.53277034956054992618397735725, 1.68122352879369004058027875935, 3.61513653101182260235750485289, 4.37635046388761817005279246450, 5.68873586166043852879053241067, 6.35779566111850847081131253576, 7.34194911718470789361483950464, 8.203141365254176962821519675612, 9.211320199864187787651749714142, 9.963238030089515835432214550898

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