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Properties

Label 2-315-63.25-c1-0-19
Degree $2$
Conductor $315$
Sign $-0.140 + 0.990i$
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  − 2.65·2-s + (1.69 − 0.354i)3-s + 5.03·4-s + (−0.5 + 0.866i)5-s + (−4.49 + 0.941i)6-s + (−1.76 − 1.96i)7-s − 8.04·8-s + (2.74 − 1.20i)9-s + (1.32 − 2.29i)10-s + (−2.66 − 4.61i)11-s + (8.53 − 1.78i)12-s + (−2.30 − 3.98i)13-s + (4.68 + 5.22i)14-s + (−0.540 + 1.64i)15-s + 11.2·16-s + (−0.923 + 1.60i)17-s + ⋯
L(s)  = 1  − 1.87·2-s + (0.978 − 0.204i)3-s + 2.51·4-s + (−0.223 + 0.387i)5-s + (−1.83 + 0.384i)6-s + (−0.667 − 0.744i)7-s − 2.84·8-s + (0.916 − 0.401i)9-s + (0.419 − 0.726i)10-s + (−0.803 − 1.39i)11-s + (2.46 − 0.515i)12-s + (−0.638 − 1.10i)13-s + (1.25 + 1.39i)14-s + (−0.139 + 0.424i)15-s + 2.81·16-s + (−0.224 + 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.140 + 0.990i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.140 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369407 - 0.425516i\)
\(L(\frac12)\) \(\approx\) \(0.369407 - 0.425516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 + 0.354i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.76 + 1.96i)T \)
good2 \( 1 + 2.65T + 2T^{2} \)
11 \( 1 + (2.66 + 4.61i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.30 + 3.98i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.923 - 1.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.39 - 2.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.852 + 1.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.11 + 3.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + (5.06 + 8.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.59 + 2.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.34 - 7.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + (-1.74 + 3.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.93T + 59T^{2} \)
61 \( 1 + 2.89T + 61T^{2} \)
67 \( 1 - 1.65T + 67T^{2} \)
71 \( 1 + 2.79T + 71T^{2} \)
73 \( 1 + (1.45 - 2.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.65T + 79T^{2} \)
83 \( 1 + (-2.26 + 3.91i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.12 - 5.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.38 - 7.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70528607314940107220223857146, −10.39968175588255588946106382491, −9.502199739926711682080913668029, −8.433692472718560581470194788337, −7.87086236353369815150174749907, −7.13808729564264484967177840070, −6.05673508050992552692154712534, −3.43158009194523617447449542306, −2.57115145917639225959135508793, −0.62963037376384132901454515078, 1.93032192100790739552791517141, 2.86874244543171297791870246465, 4.85777183879961371523158488798, 6.83513522257470236345526898930, 7.34712616191498034854603182881, 8.449609167425639491796667296588, 9.143124162249700671557497972731, 9.725438049753070667257142744159, 10.40606266990675235183125586458, 11.82734329636155525377193699178

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