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Properties

Label 2-315-9.7-c1-0-22
Degree $2$
Conductor $315$
Sign $-0.998 - 0.0497i$
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  + (1.09 − 1.88i)2-s + (−0.551 − 1.64i)3-s + (−1.37 − 2.38i)4-s + (−0.5 − 0.866i)5-s + (−3.70 − 0.748i)6-s + (0.5 − 0.866i)7-s − 1.64·8-s + (−2.39 + 1.81i)9-s − 2.18·10-s + (1.12 − 1.94i)11-s + (−3.15 + 3.57i)12-s + (2.15 + 3.73i)13-s + (−1.09 − 1.88i)14-s + (−1.14 + 1.29i)15-s + (0.959 − 1.66i)16-s − 7.61·17-s + ⋯
L(s)  = 1  + (0.770 − 1.33i)2-s + (−0.318 − 0.947i)3-s + (−0.688 − 1.19i)4-s + (−0.223 − 0.387i)5-s + (−1.51 − 0.305i)6-s + (0.188 − 0.327i)7-s − 0.582·8-s + (−0.797 + 0.603i)9-s − 0.689·10-s + (0.337 − 0.585i)11-s + (−0.911 + 1.03i)12-s + (0.597 + 1.03i)13-s + (−0.291 − 0.504i)14-s + (−0.295 + 0.335i)15-s + (0.239 − 0.415i)16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.998 - 0.0497i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.998 - 0.0497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0410517 + 1.64818i\)
\(L(\frac12)\) \(\approx\) \(0.0410517 + 1.64818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.551 + 1.64i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.09 + 1.88i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.12 + 1.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.15 - 3.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.61T + 17T^{2} \)
19 \( 1 - 2.77T + 19T^{2} \)
23 \( 1 + (-3.45 - 5.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.17 + 8.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.920 + 1.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.36T + 37T^{2} \)
41 \( 1 + (3.46 + 5.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.98 - 5.16i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.59 + 2.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.14T + 53T^{2} \)
59 \( 1 + (1.41 + 2.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.59 - 9.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.966 - 1.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.17T + 71T^{2} \)
73 \( 1 - 3.51T + 73T^{2} \)
79 \( 1 + (3.74 - 6.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + (-0.988 + 1.71i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.48990557004705931271196270829, −10.90482918108145505102215874964, −9.468381881084152683517154556827, −8.445726511299452779308884295706, −7.19410909856888201351180151786, −6.08683881242967049391560131494, −4.81788060278924122248555450825, −3.82795942939492103326111754911, −2.32535085296588022991597840837, −1.08642959768917448214648727422, 3.16060463734822689001068104319, 4.45139849130052264924278814369, 5.08982155167321907617574831474, 6.26529074241672748321967583735, 6.92862583438591568622958794288, 8.287549631147247264872518075082, 9.018689742729941195147407728281, 10.45014147795024695951245494925, 11.07701425191537870250534838776, 12.30078585125823486239459966083

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