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Properties

Label 2-795-795.158-c1-0-63
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: $\chi_{795} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 795,\ (\ :1/2),\ 0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55474 - 0.848044i\)
\(L(\frac12)\) \(\approx\) \(1.55474 - 0.848044i\)
\(L(\frac{3}{2})\) 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

−10.51960903864973578543250958014, −9.040591958970477094483114893260, −8.287468934843908867672092906529, −7.60518687478280533014443848160, −6.76336366186661090055508844895, −5.66983749896435204459539421684, −4.41966258732443911361026614110, −4.16866939542314113374388084292, −2.03725923370348122363733585486, −1.14022190531104015703679339958, 1.39493583370103720050711766792, 2.86374176722974038845086040889, 4.29694674155297186996627798942, 5.50331399140002626046639755411, 5.68965304965263240965760277573, 6.46966030550633823425008660275, 7.87621756584892870310976736361, 8.863137923237888132288112609169, 9.771478781157237817788353030156, 10.53510929955284734416391843458

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