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Properties

Label 2-1295-7.4-c1-0-1
Degree $2$
Conductor $1295$
Sign $-0.999 - 0.0259i$
Analytic cond. $10.3406$
Root an. cond. $3.21568$
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  + (0.189 + 0.328i)2-s + (−0.313 + 0.543i)3-s + (0.927 − 1.60i)4-s + (−0.5 − 0.866i)5-s − 0.238·6-s + (−2.24 + 1.40i)7-s + 1.46·8-s + (1.30 + 2.25i)9-s + (0.189 − 0.328i)10-s + (−0.195 + 0.338i)11-s + (0.582 + 1.00i)12-s − 5.61·13-s + (−0.887 − 0.469i)14-s + 0.627·15-s + (−1.57 − 2.73i)16-s + (−1.36 + 2.36i)17-s + ⋯
L(s)  = 1  + (0.134 + 0.232i)2-s + (−0.181 + 0.313i)3-s + (0.463 − 0.803i)4-s + (−0.223 − 0.387i)5-s − 0.0972·6-s + (−0.846 + 0.532i)7-s + 0.517·8-s + (0.434 + 0.752i)9-s + (0.0600 − 0.103i)10-s + (−0.0588 + 0.101i)11-s + (0.168 + 0.291i)12-s − 1.55·13-s + (−0.237 − 0.125i)14-s + 0.162·15-s + (−0.394 − 0.683i)16-s + (−0.331 + 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1295\)    =    \(5 \cdot 7 \cdot 37\)
Sign: $-0.999 - 0.0259i$
Analytic conductor: \(10.3406\)
Root analytic conductor: \(3.21568\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1295} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1295,\ (\ :1/2),\ -0.999 - 0.0259i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1351280566\)
\(L(\frac12)\) \(\approx\) \(0.1351280566\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.24 - 1.40i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.189 - 0.328i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.313 - 0.543i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.195 - 0.338i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.61T + 13T^{2} \)
17 \( 1 + (1.36 - 2.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.29 + 5.71i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.941 - 1.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.03T + 29T^{2} \)
31 \( 1 + (1.06 - 1.84i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + 9.05T + 41T^{2} \)
43 \( 1 + 5.44T + 43T^{2} \)
47 \( 1 + (0.0948 + 0.164i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.49 - 6.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.28 - 5.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.21 - 3.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.06 - 8.77i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.15T + 71T^{2} \)
73 \( 1 + (0.00378 - 0.00656i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.674 - 1.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + (2.78 + 4.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−10.07902658196190931104842288610, −9.412296025557099751201463188401, −8.532301661034706575590030700601, −7.30657777273470639084154045976, −6.84923474435280008113912240005, −5.79588327531532024622434875418, −4.99291701126385333463055644554, −4.43785383646456975178617481160, −2.81945662028868864298301605015, −1.86707278824164632289358861846, 0.04914617825468748284084184210, 1.97679101154005789031235444611, 3.10658966991797122789953752134, 3.79627063011956318002062392108, 4.78974414424570431114895980589, 6.32535712260699399629052191852, 6.80941804755324805769478970772, 7.45162399059316474972532818584, 8.225207731952509485386758985367, 9.449622651138892542766637823478

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