Properties

Label 2-315-63.38-c1-0-27
Degree $2$
Conductor $315$
Sign $-0.936 + 0.351i$
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  − 2.21i·2-s + (1.62 − 0.591i)3-s − 2.90·4-s + (−0.5 + 0.866i)5-s + (−1.30 − 3.60i)6-s + (−0.612 − 2.57i)7-s + 2.00i·8-s + (2.30 − 1.92i)9-s + (1.91 + 1.10i)10-s + (1.37 − 0.794i)11-s + (−4.72 + 1.71i)12-s + (−1.44 + 0.832i)13-s + (−5.70 + 1.35i)14-s + (−0.302 + 1.70i)15-s − 1.37·16-s + (1.59 − 2.76i)17-s + ⋯
L(s)  = 1  − 1.56i·2-s + (0.939 − 0.341i)3-s − 1.45·4-s + (−0.223 + 0.387i)5-s + (−0.534 − 1.47i)6-s + (−0.231 − 0.972i)7-s + 0.708i·8-s + (0.767 − 0.641i)9-s + (0.606 + 0.350i)10-s + (0.415 − 0.239i)11-s + (−1.36 + 0.495i)12-s + (−0.400 + 0.231i)13-s + (−1.52 + 0.362i)14-s + (−0.0780 + 0.440i)15-s − 0.343·16-s + (0.386 − 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.936 + 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.279986 - 1.54460i\)
\(L(\frac12)\) \(\approx\) \(0.279986 - 1.54460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 + 0.591i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.612 + 2.57i)T \)
good2 \( 1 + 2.21iT - 2T^{2} \)
11 \( 1 + (-1.37 + 0.794i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 - 0.832i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.59 + 2.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.82 - 3.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.37 - 4.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.79 - 2.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.44iT - 31T^{2} \)
37 \( 1 + (-5.21 - 9.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.207 + 0.358i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.14 + 3.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + (-0.301 - 0.174i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.51T + 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + 8.45iT - 71T^{2} \)
73 \( 1 + (9.85 + 5.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.04T + 79T^{2} \)
83 \( 1 + (4.33 - 7.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.218 + 0.378i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.10 + 1.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.25950286155721257336954246488, −10.31919110211499666717002452078, −9.657907555663850588630981079601, −8.726353372077582677346875575975, −7.51197128121961791382001411145, −6.63345265888219560045533617562, −4.43272536025912209381740662188, −3.57561773606095431539971254757, −2.64661060346519476645466791825, −1.16157953008436144190062944550, 2.56835198585527152921850070714, 4.27885502984955275328350125464, 5.14751016819772393679088256744, 6.38527110676377427403807757135, 7.30230314252617849590246632462, 8.480587634400270159073975976093, 8.734185328077685347940365912110, 9.668123081521611781867851982650, 10.99909303550031313951037075161, 12.59527927248102601355778135884

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