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Properties

Label 2-315-35.13-c1-0-4
Degree $2$
Conductor $315$
Sign $0.699 - 0.714i$
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  + (0.707 − 0.707i)2-s + 0.999i·4-s + 2.23i·5-s + (−2.58 − 0.581i)7-s + (2.12 + 2.12i)8-s + (1.58 + 1.58i)10-s + 4.24·11-s + (−3.16 + 3.16i)13-s + (−2.23 + 1.41i)14-s + 1.00·16-s + (2.23 + 2.23i)17-s + 3.16·19-s − 2.23·20-s + (3 − 3i)22-s + (1.41 + 1.41i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + 0.499i·4-s + 0.999i·5-s + (−0.975 − 0.219i)7-s + (0.750 + 0.750i)8-s + (0.500 + 0.500i)10-s + 1.27·11-s + (−0.877 + 0.877i)13-s + (−0.597 + 0.377i)14-s + 0.250·16-s + (0.542 + 0.542i)17-s + 0.725·19-s − 0.499·20-s + (0.639 − 0.639i)22-s + (0.294 + 0.294i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.699 - 0.714i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46446 + 0.615516i\)
\(L(\frac12)\) \(\approx\) \(1.46446 + 0.615516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 2.23iT \)
7 \( 1 + (2.58 + 0.581i)T \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (3.16 - 3.16i)T - 13iT^{2} \)
17 \( 1 + (-2.23 - 2.23i)T + 17iT^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + (-1.41 - 1.41i)T + 23iT^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 9.48iT - 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + (2 + 2i)T + 43iT^{2} \)
47 \( 1 + (-8.94 - 8.94i)T + 47iT^{2} \)
53 \( 1 + (7.07 + 7.07i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + (3.16 - 3.16i)T - 73iT^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 + (-4.47 + 4.47i)T - 83iT^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−11.81975913795963657838649913646, −11.13460353043899098208523611319, −9.938303154555188289511892742731, −9.263190521712990906080537380636, −7.69315266909373419578660458762, −6.97619642155662097234545392528, −5.95514391848715386660622723089, −4.18778371573229499211207188745, −3.49372945752301837296835472332, −2.28504892322483222189447952248, 1.09494271072425852550321804776, 3.31814760939105171016652567270, 4.73723693706320594158189786040, 5.48860295428124733982294326620, 6.52701510183870829614819722786, 7.44506820033340958612009523756, 8.913739501716608085659837608359, 9.590731259113051061489055871182, 10.38387505694155365740811198770, 11.91626329988782091659470275088

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