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Properties

Label 2-315-35.3-c1-0-4
Degree $2$
Conductor $315$
Sign $0.207 + 0.978i$
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  + (−1.49 − 0.401i)2-s + (0.346 + 0.200i)4-s + (−1.61 + 1.54i)5-s + (−2.44 + 1.01i)7-s + (1.75 + 1.75i)8-s + (3.03 − 1.66i)10-s + (2.59 − 4.49i)11-s + (3.30 − 3.30i)13-s + (4.06 − 0.545i)14-s + (−2.32 − 4.01i)16-s + (−0.0194 + 0.00519i)17-s + (−1.24 − 2.14i)19-s + (−0.869 + 0.213i)20-s + (−5.68 + 5.68i)22-s + (−0.601 + 2.24i)23-s + ⋯
L(s)  = 1  + (−1.05 − 0.283i)2-s + (0.173 + 0.100i)4-s + (−0.722 + 0.691i)5-s + (−0.922 + 0.385i)7-s + (0.619 + 0.619i)8-s + (0.960 − 0.527i)10-s + (0.782 − 1.35i)11-s + (0.917 − 0.917i)13-s + (1.08 − 0.145i)14-s + (−0.580 − 1.00i)16-s + (−0.00470 + 0.00126i)17-s + (−0.284 − 0.492i)19-s + (−0.194 + 0.0476i)20-s + (−1.21 + 1.21i)22-s + (−0.125 + 0.468i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.207 + 0.978i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.207 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387734 - 0.314220i\)
\(L(\frac12)\) \(\approx\) \(0.387734 - 0.314220i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 1.54i)T \)
7 \( 1 + (2.44 - 1.01i)T \)
good2 \( 1 + (1.49 + 0.401i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-2.59 + 4.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.30 + 3.30i)T - 13iT^{2} \)
17 \( 1 + (0.0194 - 0.00519i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.24 + 2.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.601 - 2.24i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (-5.69 - 3.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.66 - 0.714i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.68iT - 41T^{2} \)
43 \( 1 + (2.79 + 2.79i)T + 43iT^{2} \)
47 \( 1 + (0.303 - 1.13i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.60 + 1.23i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.222 - 0.385i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.18 + 0.684i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.52 + 5.70i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.14T + 71T^{2} \)
73 \( 1 + (-1.91 - 7.15i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.47 + 2.00i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.77 - 3.77i)T - 83iT^{2} \)
89 \( 1 + (-1.91 - 3.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 + 10.5i)T + 97iT^{2} \)
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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

−11.24257748473240787301873088494, −10.50089158207337879423269245465, −9.598906265207615750312963267642, −8.592778310041832939902842790758, −8.048564588846644081392950819877, −6.68745128837022359134955814022, −5.77729947986166364561300901039, −3.90938524039232401980396817470, −2.82516142141202735600699129372, −0.59820256685126850231689268834, 1.31390970717026285773296441064, 3.80954828042754052061128471096, 4.51718507488050648399041943755, 6.49618292110737643805734191488, 7.15325057232052832058607401224, 8.229908574940115488025557400276, 9.075814369838506844529515868945, 9.673704837134473452337063993914, 10.66210566587865892203741412802, 11.89100435433325624204385821447

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