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Properties

Label 2-315-63.59-c1-0-27
Degree $2$
Conductor $315$
Sign $0.0130 + 0.999i$
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  + (0.600 + 0.346i)2-s + (−0.431 − 1.67i)3-s + (−0.759 − 1.31i)4-s + 5-s + (0.322 − 1.15i)6-s + (2.55 + 0.700i)7-s − 2.43i·8-s + (−2.62 + 1.44i)9-s + (0.600 + 0.346i)10-s − 2.01i·11-s + (−1.88 + 1.84i)12-s + (−3.66 − 2.11i)13-s + (1.28 + 1.30i)14-s + (−0.431 − 1.67i)15-s + (−0.675 + 1.16i)16-s + (1.55 − 2.69i)17-s + ⋯
L(s)  = 1  + (0.424 + 0.244i)2-s + (−0.248 − 0.968i)3-s + (−0.379 − 0.658i)4-s + 0.447·5-s + (0.131 − 0.471i)6-s + (0.964 + 0.264i)7-s − 0.862i·8-s + (−0.876 + 0.482i)9-s + (0.189 + 0.109i)10-s − 0.608i·11-s + (−0.542 + 0.531i)12-s + (−1.01 − 0.586i)13-s + (0.344 + 0.348i)14-s + (−0.111 − 0.433i)15-s + (−0.168 + 0.292i)16-s + (0.376 − 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0130 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0130 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0130 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0130 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03661 - 1.02319i\)
\(L(\frac12)\) \(\approx\) \(1.03661 - 1.02319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.431 + 1.67i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.55 - 0.700i)T \)
good2 \( 1 + (-0.600 - 0.346i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 2.01iT - 11T^{2} \)
13 \( 1 + (3.66 + 2.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.55 + 2.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.112 - 0.0652i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.377iT - 23T^{2} \)
29 \( 1 + (-7.55 + 4.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.12 - 1.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.94 - 5.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.02 + 1.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.79 - 8.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.31 - 4.00i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.28 - 4.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.68 + 9.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.64 - 2.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.76 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.48iT - 71T^{2} \)
73 \( 1 + (1.34 + 0.777i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.35 + 2.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.308 - 0.534i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.725 + 1.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.3 - 6.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.58102131375899856746427657896, −10.59820851903043940643196458622, −9.573850797535242385280645128931, −8.420773591859223528964810147781, −7.49074015066413602390577096112, −6.32395863304346516282118830267, −5.48591733579665613240712857823, −4.75104102675855086404940144829, −2.63235134714309730741535957602, −1.04458471276585041078153194771, 2.36217862419800618140648887416, 3.90022573051531626766940213428, 4.70618119461151858471603048673, 5.47455228290123085152726111320, 7.11333144848263135666338632281, 8.275591013792329997966287171418, 9.156353795901361904941146687044, 10.12920854347081753399767683809, 10.99176039466218303035765206111, 11.95903112358724415211219320580

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