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Properties

Label 2-315-35.33-c1-0-1
Degree $2$
Conductor $315$
Sign $-0.992 + 0.121i$
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  + (0.582 + 2.17i)2-s + (−2.65 + 1.53i)4-s + (0.0591 + 2.23i)5-s + (−2.56 + 0.660i)7-s + (−1.69 − 1.69i)8-s + (−4.82 + 1.43i)10-s + (−0.329 − 0.571i)11-s + (2.55 − 2.55i)13-s + (−2.92 − 5.18i)14-s + (−0.366 + 0.635i)16-s + (−1.45 + 5.43i)17-s + (1.48 − 2.56i)19-s + (−3.58 − 5.84i)20-s + (1.04 − 1.04i)22-s + (0.271 − 0.0726i)23-s + ⋯
L(s)  = 1  + (0.411 + 1.53i)2-s + (−1.32 + 0.766i)4-s + (0.0264 + 0.999i)5-s + (−0.968 + 0.249i)7-s + (−0.599 − 0.599i)8-s + (−1.52 + 0.452i)10-s + (−0.0994 − 0.172i)11-s + (0.709 − 0.709i)13-s + (−0.782 − 1.38i)14-s + (−0.0916 + 0.158i)16-s + (−0.353 + 1.31i)17-s + (0.339 − 0.588i)19-s + (−0.801 − 1.30i)20-s + (0.223 − 0.223i)22-s + (0.0565 − 0.0151i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.992 + 0.121i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.992 + 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0803622 - 1.31640i\)
\(L(\frac12)\) \(\approx\) \(0.0803622 - 1.31640i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0591 - 2.23i)T \)
7 \( 1 + (2.56 - 0.660i)T \)
good2 \( 1 + (-0.582 - 2.17i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.329 + 0.571i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.55 + 2.55i)T - 13iT^{2} \)
17 \( 1 + (1.45 - 5.43i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.48 + 2.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.271 + 0.0726i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.03iT - 29T^{2} \)
31 \( 1 + (-6.53 + 3.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.08 - 7.79i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (-8.53 - 8.53i)T + 43iT^{2} \)
47 \( 1 + (-11.6 + 3.11i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.24 - 4.65i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.782 + 1.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.02 + 0.589i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.84 + 0.762i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.13T + 71T^{2} \)
73 \( 1 + (-1.55 - 0.417i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.17 + 3.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.14 - 2.14i)T - 83iT^{2} \)
89 \( 1 + (2.24 - 3.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.33 - 3.33i)T + 97iT^{2} \)
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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

−12.51651717798943022564076691171, −11.07303569299750075383181265811, −10.26547069703929310591578642752, −9.009740357714608523430070133814, −8.070453429989935073173895368687, −7.09005387866791577401407757622, −6.24575275534795837186313564896, −5.74095866189544913168272092935, −4.14046276611571927099453137376, −2.94834678887823307166644720095, 0.879678783174033022308860252785, 2.47532702066199384665986015179, 3.81110211664737278098598964506, 4.62297485224086913248993618704, 5.91612883714100415019802626108, 7.34521926987930817801126672138, 8.902543186176695954995676178677, 9.502437630949400847383151141798, 10.30797943411344383983940758100, 11.40653176966761422800290198587

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