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Properties

Label 2-315-35.33-c1-0-8
Degree $2$
Conductor $315$
Sign $0.00446 - 0.999i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (0.461 + 1.72i)2-s + (−1.01 + 0.587i)4-s + (1.92 + 1.14i)5-s + (2.04 − 1.68i)7-s + (1.03 + 1.03i)8-s + (−1.07 + 3.83i)10-s + (−1.46 − 2.54i)11-s + (0.187 − 0.187i)13-s + (3.83 + 2.74i)14-s + (−2.48 + 4.30i)16-s + (0.868 − 3.24i)17-s + (−1.81 + 3.14i)19-s + (−2.62 − 0.0328i)20-s + (3.69 − 3.69i)22-s + (−9.07 + 2.43i)23-s + ⋯
L(s)  = 1  + (0.326 + 1.21i)2-s + (−0.508 + 0.293i)4-s + (0.859 + 0.510i)5-s + (0.772 − 0.635i)7-s + (0.367 + 0.367i)8-s + (−0.341 + 1.21i)10-s + (−0.442 − 0.766i)11-s + (0.0521 − 0.0521i)13-s + (1.02 + 0.732i)14-s + (−0.621 + 1.07i)16-s + (0.210 − 0.786i)17-s + (−0.417 + 0.722i)19-s + (−0.587 − 0.00734i)20-s + (0.788 − 0.788i)22-s + (−1.89 + 0.507i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00446 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00446 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.00446 - 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.00446 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34496 + 1.33897i\)
\(L(\frac12)\) \(\approx\) \(1.34496 + 1.33897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.92 - 1.14i)T \)
7 \( 1 + (-2.04 + 1.68i)T \)
good2 \( 1 + (-0.461 - 1.72i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.46 + 2.54i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.187 + 0.187i)T - 13iT^{2} \)
17 \( 1 + (-0.868 + 3.24i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.81 - 3.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (9.07 - 2.43i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.815iT - 29T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.69 + 6.31i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 + (3.59 + 3.59i)T + 43iT^{2} \)
47 \( 1 + (-9.24 + 2.47i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.30 + 4.87i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.41 - 9.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.07 - 4.66i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (15.3 + 4.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.68T + 71T^{2} \)
73 \( 1 + (-1.89 - 0.508i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.44 - 2.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.09 + 6.09i)T - 83iT^{2} \)
89 \( 1 + (4.87 - 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.93 - 5.93i)T + 97iT^{2} \)
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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

−11.84537215727130433500156415408, −10.76928523010205609034276220484, −10.19264396093835791040634725404, −8.755205986484097258535737044795, −7.77391166335113391804340980359, −7.06529934852559450972831527751, −5.88371233952591663586691639452, −5.36594075390501842585992746684, −3.91520151851405969791460722208, −2.02835224789395533440955724039, 1.70026308220258230784087556446, 2.46695261238329638840013857995, 4.20452242917701261253177439339, 5.09547139222729894783162780904, 6.28373397432791831483884564453, 7.79945563499856093563887488163, 8.843880336507437492717444126939, 9.903020649361810109548413093796, 10.50955806930863602649986867234, 11.55523851701650434950220508058

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