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Properties

Label 2-315-315.34-c0-0-1
Degree $2$
Conductor $315$
Sign $0.766 + 0.642i$
Analytic cond. $0.157205$
Root an. cond. $0.396491$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 9-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 + 0.866i)16-s − 17-s + (−0.499 + 0.866i)20-s + (−0.5 + 0.866i)21-s + (−0.499 + 0.866i)25-s + 27-s + 0.999·28-s + ⋯
L(s)  = 1  + 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 9-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 + 0.866i)16-s − 17-s + (−0.499 + 0.866i)20-s + (−0.5 + 0.866i)21-s + (−0.499 + 0.866i)25-s + 27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.157205\)
Root analytic conductor: \(0.396491\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8777030976\)
\(L(\frac12)\) \(\approx\) \(0.8777030976\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.83881925072780939452790170688, −10.80856202486761247304944346622, −9.456569191986250616637180472093, −8.889002196645706859135933612168, −8.574958860136773651939128232757, −6.94005287664058914355889217412, −5.79368019381748874088411136543, −4.57848228841149103640878300130, −3.53913634701750426091528981135, −1.70821105391874785835466218405, 2.63303018824714148227077228516, 3.76475874034298914605932328799, 4.28595429585172042586611837020, 6.56583961080603925279781164907, 7.44906671391731656241861529965, 7.998381386217842099469961426020, 9.170630226804899325818910105422, 9.977854288867507135827628902561, 10.97901009140078801521546686572, 12.14934561063154958854333425236

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