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Properties

Label 2-795-795.317-c1-0-81
Degree $2$
Conductor $795$
Sign $-0.541 + 0.840i$
Analytic cond. $6.34810$
Root an. cond. $2.51954$
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  + (−0.278 − 0.278i)2-s + (1.22 − 1.22i)3-s − 1.84i·4-s + (−0.962 − 2.01i)5-s − 0.682·6-s + (3.42 + 3.42i)7-s + (−1.07 + 1.07i)8-s − 2.99i·9-s + (−0.294 + 0.830i)10-s + (−2.25 − 2.25i)12-s + (5.09 − 5.09i)13-s − 1.91i·14-s + (−3.65 − 1.29i)15-s − 3.09·16-s + (−0.836 + 0.836i)18-s + ⋯
L(s)  = 1  + (−0.197 − 0.197i)2-s + (0.707 − 0.707i)3-s − 0.922i·4-s + (−0.430 − 0.902i)5-s − 0.278·6-s + (1.29 + 1.29i)7-s + (−0.378 + 0.378i)8-s − 0.999i·9-s + (−0.0930 + 0.262i)10-s + (−0.652 − 0.652i)12-s + (1.41 − 1.41i)13-s − 0.510i·14-s + (−0.942 − 0.333i)15-s − 0.773·16-s + (−0.197 + 0.197i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(795\)    =    \(3 \cdot 5 \cdot 53\)
Sign: $-0.541 + 0.840i$
Analytic conductor: \(6.34810\)
Root analytic conductor: \(2.51954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{795} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 795,\ (\ :1/2),\ -0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893746 - 1.63853i\)
\(L(\frac12)\) \(\approx\) \(0.893746 - 1.63853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (0.962 + 2.01i)T \)
53 \( 1 + (5.14 - 5.14i)T \)
good2 \( 1 + (0.278 + 0.278i)T + 2iT^{2} \)
7 \( 1 + (-3.42 - 3.42i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-5.09 + 5.09i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-1.59 + 1.59i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-4.03 - 4.03i)T + 37iT^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 + (4.95 - 4.95i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 16.1T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-4.14 + 4.14i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (9.88 + 9.88i)T + 97iT^{2} \)
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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

−9.797150385955404470635388852959, −8.782678340323995625388442043169, −8.459548553303518704707594713479, −7.86798144359255679511050570855, −6.32557756956705457067034123816, −5.53655793131697389173154540259, −4.75593946197302173934046000694, −3.16823176255189050241278365402, −1.85813549134398871283535280316, −1.02503322901790325358678093700, 1.93086234382965387346065830155, 3.52558539886320846882491210248, 3.88854316435496612843582320795, 4.79086196074200983629524676971, 6.64567236778367154085617925844, 7.28715484413848987699934803104, 8.122272889604856198759094305558, 8.575829074039863523412493635430, 9.654128647286726200130187603318, 10.72319887700231676495212612375

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