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Properties

Label 2-795-795.314-c0-0-1
Degree $2$
Conductor $795$
Sign $0.0693 + 0.997i$
Analytic cond. $0.396756$
Root an. cond. $0.629886$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.213 − 1.75i)2-s + (0.885 + 0.464i)3-s + (−2.07 − 0.511i)4-s + (0.568 + 0.822i)5-s + (1.00 − 1.45i)6-s + (−0.713 + 1.88i)8-s + (0.568 + 0.822i)9-s + (1.56 − 0.822i)10-s + (−1.59 − 1.41i)12-s + (0.120 + 0.992i)15-s + (1.26 + 0.663i)16-s + (−0.627 − 1.65i)17-s + (1.56 − 0.822i)18-s + (1.45 − 0.358i)19-s + (−0.757 − 1.99i)20-s + ⋯
L(s)  = 1  + (0.213 − 1.75i)2-s + (0.885 + 0.464i)3-s + (−2.07 − 0.511i)4-s + (0.568 + 0.822i)5-s + (1.00 − 1.45i)6-s + (−0.713 + 1.88i)8-s + (0.568 + 0.822i)9-s + (1.56 − 0.822i)10-s + (−1.59 − 1.41i)12-s + (0.120 + 0.992i)15-s + (1.26 + 0.663i)16-s + (−0.627 − 1.65i)17-s + (1.56 − 0.822i)18-s + (1.45 − 0.358i)19-s + (−0.757 − 1.99i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0693 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 795 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0693 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(795\)    =    \(3 \cdot 5 \cdot 53\)
Sign: $0.0693 + 0.997i$
Analytic conductor: \(0.396756\)
Root analytic conductor: \(0.629886\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{795} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 795,\ (\ :0),\ 0.0693 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368798340\)
\(L(\frac12)\) \(\approx\) \(1.368798340\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.885 - 0.464i)T \)
5 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
good2 \( 1 + (-0.213 + 1.75i)T + (-0.970 - 0.239i)T^{2} \)
7 \( 1 + (0.970 + 0.239i)T^{2} \)
11 \( 1 + (-0.120 - 0.992i)T^{2} \)
13 \( 1 + (-0.885 + 0.464i)T^{2} \)
17 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
19 \( 1 + (-1.45 + 0.358i)T + (0.885 - 0.464i)T^{2} \)
23 \( 1 + 1.49T + T^{2} \)
29 \( 1 + (-0.120 + 0.992i)T^{2} \)
31 \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \)
37 \( 1 + (-0.568 - 0.822i)T^{2} \)
41 \( 1 + (-0.120 - 0.992i)T^{2} \)
43 \( 1 + (-0.568 + 0.822i)T^{2} \)
47 \( 1 + (1.10 - 1.59i)T + (-0.354 - 0.935i)T^{2} \)
59 \( 1 + (0.354 + 0.935i)T^{2} \)
61 \( 1 + (-0.251 + 0.663i)T + (-0.748 - 0.663i)T^{2} \)
67 \( 1 + (-0.885 - 0.464i)T^{2} \)
71 \( 1 + (-0.568 + 0.822i)T^{2} \)
73 \( 1 + (0.748 - 0.663i)T^{2} \)
79 \( 1 + (-0.136 - 1.12i)T + (-0.970 + 0.239i)T^{2} \)
83 \( 1 + 1.94T + T^{2} \)
89 \( 1 + (0.748 - 0.663i)T^{2} \)
97 \( 1 + (0.354 - 0.935i)T^{2} \)
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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

−10.14197444893874394423262134334, −9.672441578881173529988130733139, −9.222068173765459964392276442645, −7.994347200863883853053674404910, −6.92099247344122725941766622667, −5.36128363622360579445364785416, −4.48202302448309188741588801360, −3.32920635794179642246734075487, −2.77680880625161882553669459018, −1.80210300311186046057342541103, 1.77294826242137563148361626168, 3.66580255215703646681362163873, 4.56723057705579056550951890763, 5.73915579046719287798202350238, 6.27988946736297455974971585660, 7.31180369542408422095913853594, 8.158737885131747292923773171391, 8.536149308412370518686936139010, 9.435245452829576750111273350905, 10.10917953602794165809854608943

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