Properties

Label 2-315-63.5-c1-0-11
Degree $2$
Conductor $315$
Sign $0.749 + 0.662i$
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  − 2.34i·2-s + (0.822 + 1.52i)3-s − 3.50·4-s + (0.5 + 0.866i)5-s + (3.57 − 1.92i)6-s + (2.00 + 1.73i)7-s + 3.53i·8-s + (−1.64 + 2.50i)9-s + (2.03 − 1.17i)10-s + (4.81 + 2.77i)11-s + (−2.88 − 5.34i)12-s + (−0.0395 − 0.0228i)13-s + (4.06 − 4.69i)14-s + (−0.909 + 1.47i)15-s + 1.27·16-s + (−0.320 − 0.554i)17-s + ⋯
L(s)  = 1  − 1.65i·2-s + (0.474 + 0.880i)3-s − 1.75·4-s + (0.223 + 0.387i)5-s + (1.46 − 0.787i)6-s + (0.756 + 0.654i)7-s + 1.24i·8-s + (−0.549 + 0.835i)9-s + (0.642 − 0.370i)10-s + (1.45 + 0.837i)11-s + (−0.831 − 1.54i)12-s + (−0.0109 − 0.00632i)13-s + (1.08 − 1.25i)14-s + (−0.234 + 0.380i)15-s + 0.319·16-s + (−0.0776 − 0.134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.749 + 0.662i$
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: \((2,\ 315,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46651 - 0.554974i\)
\(L(\frac12)\) \(\approx\) \(1.46651 - 0.554974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.822 - 1.52i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.00 - 1.73i)T \)
good2 \( 1 + 2.34iT - 2T^{2} \)
11 \( 1 + (-4.81 - 2.77i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0395 + 0.0228i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.320 + 0.554i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.73 + 3.31i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.23 + 3.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.95 + 3.44i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.48iT - 31T^{2} \)
37 \( 1 + (3.67 - 6.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.580 + 1.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.08 + 3.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.58T + 47T^{2} \)
53 \( 1 + (5.08 - 2.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.95T + 59T^{2} \)
61 \( 1 + 6.68iT - 61T^{2} \)
67 \( 1 - 3.50T + 67T^{2} \)
71 \( 1 - 7.67iT - 71T^{2} \)
73 \( 1 + (3.76 - 2.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.56T + 79T^{2} \)
83 \( 1 + (0.725 + 1.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.24 + 16.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.49 - 3.17i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.40140058577463969286639869999, −10.73519383541330560325980404824, −9.807417503492046000179465683633, −9.120072162967858938398298586795, −8.397770520097214199435183336448, −6.60515735486917264952719392432, −4.82205059838901133690707560995, −4.21020161720309535271544369753, −2.85095822063703388223044238345, −1.90796222271734005967212710166, 1.35250017543434240251532144975, 3.77137082179797957167275771570, 5.05547359644966268316271109243, 6.31422711569956925438119801222, 6.81462165079297888593993643266, 7.915064142217505852913365166692, 8.605943731715949494340550510443, 9.181010738566086586256045271725, 10.85455908770058229341460147497, 12.02248721898896014571334108936

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