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Properties

Label 2-315-63.41-c1-0-29
Degree $2$
Conductor $315$
Sign $0.195 + 0.980i$
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  + (1.99 − 1.15i)2-s + (1.43 − 0.965i)3-s + (1.66 − 2.88i)4-s + (−0.5 + 0.866i)5-s + (1.76 − 3.58i)6-s + (−2.42 + 1.05i)7-s − 3.06i·8-s + (1.13 − 2.77i)9-s + 2.30i·10-s + (−0.565 + 0.326i)11-s + (−0.387 − 5.74i)12-s + (−0.0750 − 0.0433i)13-s + (−3.62 + 4.91i)14-s + (0.116 + 1.72i)15-s + (−0.207 − 0.358i)16-s + 0.482·17-s + ⋯
L(s)  = 1  + (1.41 − 0.815i)2-s + (0.830 − 0.557i)3-s + (0.831 − 1.44i)4-s + (−0.223 + 0.387i)5-s + (0.719 − 1.46i)6-s + (−0.916 + 0.399i)7-s − 1.08i·8-s + (0.379 − 0.925i)9-s + 0.729i·10-s + (−0.170 + 0.0985i)11-s + (−0.111 − 1.65i)12-s + (−0.0208 − 0.0120i)13-s + (−0.969 + 1.31i)14-s + (0.0300 + 0.446i)15-s + (−0.0517 − 0.0896i)16-s + 0.116·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35521 - 1.93170i\)
\(L(\frac12)\) \(\approx\) \(2.35521 - 1.93170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.43 + 0.965i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.42 - 1.05i)T \)
good2 \( 1 + (-1.99 + 1.15i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (0.565 - 0.326i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0750 + 0.0433i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.482T + 17T^{2} \)
19 \( 1 - 2.08iT - 19T^{2} \)
23 \( 1 + (-3.12 - 1.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.04 - 4.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.47 - 0.853i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + (-4.19 + 7.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.84 - 8.38i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.00 + 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.22iT - 53T^{2} \)
59 \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.21 - 3.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.729 + 1.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.97iT - 71T^{2} \)
73 \( 1 - 8.43iT - 73T^{2} \)
79 \( 1 + (5.08 + 8.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.998 - 1.72i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (14.6 - 8.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.87027760037774688296244682192, −10.78016633865397049156486276999, −9.781426233989711606945598469176, −8.716575468234521519207639755921, −7.36161738359322086851132248577, −6.41154696831878055107161166788, −5.31612848652657857521758447969, −3.73350261176266636223648730960, −3.15379567084983473518995302337, −1.97728503168849326452501465773, 2.86778698222047996982809824207, 3.82082122382650372739217783368, 4.66790868102608985588392167318, 5.75906362422138534307494387947, 6.97661005839128512148610740016, 7.75130958911233838361724030947, 8.944122836420615682278851607448, 9.874052366423934937979080861348, 11.04651099095209321377557533871, 12.37264310982615238606030852005

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