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Properties

Label 2-315-9.4-c1-0-12
Degree $2$
Conductor $315$
Sign $0.559 + 0.829i$
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  + (−0.104 − 0.181i)2-s + (−1.28 + 1.15i)3-s + (0.978 − 1.69i)4-s + (0.5 − 0.866i)5-s + (0.344 + 0.111i)6-s + (0.5 + 0.866i)7-s − 0.827·8-s + (0.313 − 2.98i)9-s − 0.209·10-s + (−1.10 − 1.91i)11-s + (0.704 + 3.31i)12-s + (1.24 − 2.15i)13-s + (0.104 − 0.181i)14-s + (0.360 + 1.69i)15-s + (−1.86 − 3.23i)16-s − 0.209·17-s + ⋯
L(s)  = 1  + (−0.0739 − 0.128i)2-s + (−0.743 + 0.669i)3-s + (0.489 − 0.847i)4-s + (0.223 − 0.387i)5-s + (0.140 + 0.0456i)6-s + (0.188 + 0.327i)7-s − 0.292·8-s + (0.104 − 0.994i)9-s − 0.0661·10-s + (−0.333 − 0.576i)11-s + (0.203 + 0.956i)12-s + (0.345 − 0.597i)13-s + (0.0279 − 0.0483i)14-s + (0.0929 + 0.437i)15-s + (−0.467 − 0.809i)16-s − 0.0507·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.559 + 0.829i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.559 + 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.991689 - 0.527290i\)
\(L(\frac12)\) \(\approx\) \(0.991689 - 0.527290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.15i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.104 + 0.181i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.10 + 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.24 + 2.15i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.209T + 17T^{2} \)
19 \( 1 - 7.22T + 19T^{2} \)
23 \( 1 + (-1.42 + 2.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.41 + 4.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.99 + 3.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.739T + 37T^{2} \)
41 \( 1 + (1.09 - 1.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.65 - 4.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.30 - 7.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 + (7.01 - 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.206 + 0.358i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.19 + 3.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.07T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + (-7.86 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.56 + 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + (-5.95 - 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.36797613384325928143040966881, −10.65633756531329121969206158989, −9.774942014689033616056631309039, −9.063275058505878020253203243377, −7.65804265711714437414284006239, −6.14779087212772308197453482748, −5.65391409236178179862385760411, −4.68344626265241798834235830247, −2.96694711042788266459235065970, −0.983294803129153797982880801605, 1.80148891083579582944791580404, 3.30963002242333327822755348794, 4.91128189222529415259664333361, 6.12966781286893159812956049070, 7.24960083763396306983144473863, 7.46060136261952389061750082378, 8.831238873997059693759222440959, 10.16018496796443203332307965787, 11.14624368130088898322164254765, 11.74985556888369950449030292416

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