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Properties

Label 2-315-63.20-c1-0-1
Degree $2$
Conductor $315$
Sign $0.316 - 0.948i$
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  + (−1.13 − 0.653i)2-s + (0.957 + 1.44i)3-s + (−0.145 − 0.252i)4-s + (−0.5 − 0.866i)5-s + (−0.140 − 2.25i)6-s + (−2.56 + 0.659i)7-s + 2.99i·8-s + (−1.16 + 2.76i)9-s + 1.30i·10-s + (4.98 + 2.88i)11-s + (0.224 − 0.452i)12-s + (−3.14 + 1.81i)13-s + (3.33 + 0.927i)14-s + (0.771 − 1.55i)15-s + (1.66 − 2.88i)16-s + 4.92·17-s + ⋯
L(s)  = 1  + (−0.800 − 0.462i)2-s + (0.552 + 0.833i)3-s + (−0.0729 − 0.126i)4-s + (−0.223 − 0.387i)5-s + (−0.0574 − 0.922i)6-s + (−0.968 + 0.249i)7-s + 1.05i·8-s + (−0.388 + 0.921i)9-s + 0.413i·10-s + (1.50 + 0.868i)11-s + (0.0649 − 0.130i)12-s + (−0.871 + 0.502i)13-s + (0.890 + 0.248i)14-s + (0.199 − 0.400i)15-s + (0.416 − 0.721i)16-s + 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599255 + 0.431852i\)
\(L(\frac12)\) \(\approx\) \(0.599255 + 0.431852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.957 - 1.44i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.56 - 0.659i)T \)
good2 \( 1 + (1.13 + 0.653i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-4.98 - 2.88i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.14 - 1.81i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.92T + 17T^{2} \)
19 \( 1 - 3.02iT - 19T^{2} \)
23 \( 1 + (5.93 - 3.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.16 - 4.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.18 - 1.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.80T + 37T^{2} \)
41 \( 1 + (5.57 + 9.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.136 - 0.235i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.88 - 3.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.51iT - 53T^{2} \)
59 \( 1 + (-4.33 - 7.51i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.81 + 4.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.65 - 2.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.95iT - 71T^{2} \)
73 \( 1 + 2.88iT - 73T^{2} \)
79 \( 1 + (-5.15 + 8.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.69 - 2.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.68T + 89T^{2} \)
97 \( 1 + (4.00 + 2.31i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.95341145572744980677058678055, −10.44836057318138482131068192575, −9.747193688746193043485399709224, −9.369735246715375257205778887950, −8.508469186727352717991085694570, −7.33477992817329361391612364120, −5.80437329204398963908166595588, −4.58552178292872404484449847713, −3.43844420047900650517278613562, −1.82190607097378613962497751387, 0.67314576095302770219733325656, 2.98300018540394812401892413554, 3.87901807953094035178349152422, 6.24431900452195522373665411308, 6.74347226815948792310050533813, 7.75937357270889677897551143183, 8.447184811267369001176548734840, 9.493382456561559348030927621165, 10.06052513114060138674621323001, 11.73223807188223436098039747015

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