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Properties

Label 2-315-63.5-c1-0-23
Degree $2$
Conductor $315$
Sign $-0.320 + 0.947i$
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  − 1.23i·2-s + (0.460 − 1.66i)3-s + 0.473·4-s + (0.5 + 0.866i)5-s + (−2.06 − 0.568i)6-s + (2.47 + 0.930i)7-s − 3.05i·8-s + (−2.57 − 1.53i)9-s + (1.07 − 0.617i)10-s + (−1.10 − 0.635i)11-s + (0.217 − 0.789i)12-s + (1.67 + 0.968i)13-s + (1.15 − 3.06i)14-s + (1.67 − 0.436i)15-s − 2.83·16-s + (0.676 + 1.17i)17-s + ⋯
L(s)  = 1  − 0.873i·2-s + (0.265 − 0.964i)3-s + 0.236·4-s + (0.223 + 0.387i)5-s + (−0.842 − 0.232i)6-s + (0.936 + 0.351i)7-s − 1.08i·8-s + (−0.858 − 0.512i)9-s + (0.338 − 0.195i)10-s + (−0.331 − 0.191i)11-s + (0.0628 − 0.228i)12-s + (0.465 + 0.268i)13-s + (0.307 − 0.817i)14-s + (0.432 − 0.112i)15-s − 0.707·16-s + (0.164 + 0.284i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.320 + 0.947i$
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: \((2,\ 315,\ (\ :1/2),\ -0.320 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00538 - 1.40141i\)
\(L(\frac12)\) \(\approx\) \(1.00538 - 1.40141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.460 + 1.66i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.47 - 0.930i)T \)
good2 \( 1 + 1.23iT - 2T^{2} \)
11 \( 1 + (1.10 + 0.635i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.67 - 0.968i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.676 - 1.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.724 + 0.418i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.914 - 0.527i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.49 - 4.90i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.24iT - 31T^{2} \)
37 \( 1 + (4.38 - 7.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.63 - 6.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.69T + 47T^{2} \)
53 \( 1 + (0.148 - 0.0855i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.77T + 59T^{2} \)
61 \( 1 + 2.84iT - 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 6.48iT - 71T^{2} \)
73 \( 1 + (-9.10 + 5.25i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.0T + 79T^{2} \)
83 \( 1 + (-4.56 - 7.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.41 - 16.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.14 - 4.70i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.28247313170118377952363721942, −10.92024114941432236783134248989, −9.598170376273395239054721855954, −8.511816041411763793114983364225, −7.55076526958771952805336253264, −6.59447857353884153614161915522, −5.52773943221756159095601567742, −3.66998922065598058328366207112, −2.43821510309469655871574256284, −1.49764573185984965076356241414, 2.22156652903463851028744772938, 3.97067019266658732690646995274, 5.15212352863245545170433957633, 5.79190362004463300819844771098, 7.31451546678871095284189864294, 8.127023337021119327919031744624, 8.887125511410220056041949624983, 10.07309739295430367753512697233, 10.98424157567321345679122496936, 11.61559342180645066162352945622

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