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Properties

Label 2-315-63.5-c1-0-3
Degree $2$
Conductor $315$
Sign $-0.230 + 0.973i$
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  + 2.80i·2-s + (0.0857 + 1.72i)3-s − 5.88·4-s + (0.5 + 0.866i)5-s + (−4.85 + 0.240i)6-s + (1.63 + 2.07i)7-s − 10.9i·8-s + (−2.98 + 0.296i)9-s + (−2.43 + 1.40i)10-s + (−2.55 − 1.47i)11-s + (−0.504 − 10.1i)12-s + (1.90 + 1.10i)13-s + (−5.83 + 4.60i)14-s + (−1.45 + 0.939i)15-s + 18.8·16-s + (1.11 + 1.93i)17-s + ⋯
L(s)  = 1  + 1.98i·2-s + (0.0495 + 0.998i)3-s − 2.94·4-s + (0.223 + 0.387i)5-s + (−1.98 + 0.0983i)6-s + (0.619 + 0.785i)7-s − 3.85i·8-s + (−0.995 + 0.0989i)9-s + (−0.768 + 0.443i)10-s + (−0.770 − 0.444i)11-s + (−0.145 − 2.93i)12-s + (0.529 + 0.305i)13-s + (−1.55 + 1.22i)14-s + (−0.375 + 0.242i)15-s + 4.70·16-s + (0.270 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.230 + 0.973i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631374 - 0.798186i\)
\(L(\frac12)\) \(\approx\) \(0.631374 - 0.798186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0857 - 1.72i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.63 - 2.07i)T \)
good2 \( 1 - 2.80iT - 2T^{2} \)
11 \( 1 + (2.55 + 1.47i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.90 - 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.11 - 1.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.10 - 1.21i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.142 + 0.0820i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.75 - 1.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.65iT - 31T^{2} \)
37 \( 1 + (-0.761 + 1.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.32 + 5.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.57 - 7.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + (-2.54 + 1.47i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.24T + 59T^{2} \)
61 \( 1 + 7.13iT - 61T^{2} \)
67 \( 1 + 3.36T + 67T^{2} \)
71 \( 1 - 6.49iT - 71T^{2} \)
73 \( 1 + (-3.71 + 2.14i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.286T + 79T^{2} \)
83 \( 1 + (-2.67 - 4.62i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.937 + 1.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 + 5.80i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.66084680288457493142403824502, −11.15608144775538707581191431214, −10.05786731256850934859274037869, −9.169730048034408789903288766896, −8.415322777468692352361278549217, −7.68542815470806030282002775585, −6.21914552015872793929007391023, −5.57627136583241110708859236330, −4.74636478100236438034428135586, −3.45870487029754810301773988774, 0.792686877273743865980395687771, 1.95159897334659534069076826712, 3.18601533736479896725707292533, 4.59553684116882272942427134895, 5.56249753382231791684036135789, 7.58463540925731308452006180980, 8.326541263948324100003158429461, 9.369334389736934429242587679297, 10.33824146545187200085341427048, 11.17485832923522940345069557193

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