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Properties

Label 2-315-63.41-c1-0-6
Degree $2$
Conductor $315$
Sign $0.812 - 0.582i$
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  + (−1.41 + 0.817i)2-s + (−1.73 + 0.0392i)3-s + (0.335 − 0.581i)4-s + (0.5 − 0.866i)5-s + (2.41 − 1.47i)6-s + (−2.43 + 1.03i)7-s − 2.17i·8-s + (2.99 − 0.135i)9-s + 1.63i·10-s + (1.25 − 0.727i)11-s + (−0.558 + 1.02i)12-s + (−4.17 − 2.40i)13-s + (2.59 − 3.45i)14-s + (−0.831 + 1.51i)15-s + (2.44 + 4.23i)16-s + 6.37·17-s + ⋯
L(s)  = 1  + (−1.00 + 0.577i)2-s + (−0.999 + 0.0226i)3-s + (0.167 − 0.290i)4-s + (0.223 − 0.387i)5-s + (0.987 − 0.600i)6-s + (−0.920 + 0.391i)7-s − 0.767i·8-s + (0.998 − 0.0453i)9-s + 0.516i·10-s + (0.379 − 0.219i)11-s + (−0.161 + 0.294i)12-s + (−1.15 − 0.667i)13-s + (0.694 − 0.923i)14-s + (−0.214 + 0.392i)15-s + (0.611 + 1.05i)16-s + 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.812 - 0.582i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.812 - 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471312 + 0.151523i\)
\(L(\frac12)\) \(\approx\) \(0.471312 + 0.151523i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0392i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.43 - 1.03i)T \)
good2 \( 1 + (1.41 - 0.817i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-1.25 + 0.727i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.17 + 2.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.37T + 17T^{2} \)
19 \( 1 - 1.12iT - 19T^{2} \)
23 \( 1 + (-7.62 - 4.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.63 - 2.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.94 - 4.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.26T + 37T^{2} \)
41 \( 1 + (-4.21 + 7.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 - 5.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.739 + 1.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.46iT - 53T^{2} \)
59 \( 1 + (1.63 - 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.44 + 3.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.282 - 0.489i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.7iT - 71T^{2} \)
73 \( 1 - 6.61iT - 73T^{2} \)
79 \( 1 + (3.80 + 6.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.62 - 2.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.18T + 89T^{2} \)
97 \( 1 + (-10.0 + 5.79i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.83832130653808253646535023283, −10.44162224603531258855868211583, −9.738131144973666275749883559761, −9.158966837121289202875072390732, −7.80901365048513042788298480632, −7.02608558176879316909355229038, −5.99583895306399501062022877055, −5.10829597678647380736745020939, −3.40106334369086414307379846782, −0.868674160002360936869013418212, 0.897871452008684835929230705416, 2.68007210697891825965522877393, 4.48902066562425897091380714812, 5.74212520195348291750259208725, 6.82782827358492822649781984110, 7.65180231168056999450918143504, 9.316721730275194313737986064993, 9.802704788142026762965455837462, 10.46769090064371983338439245299, 11.38978406658701234405948441832

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