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Properties

Label 2-315-35.33-c1-0-10
Degree $2$
Conductor $315$
Sign $0.995 - 0.0932i$
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  + (0.133 + 0.5i)2-s + (1.5 − 0.866i)4-s + (1.86 − 1.23i)5-s + (−0.866 + 2.5i)7-s + (1.36 + 1.36i)8-s + (0.866 + 0.767i)10-s + (−1.36 − 2.36i)11-s + (2 − 2i)13-s + (−1.36 − 0.0980i)14-s + (1.23 − 2.13i)16-s + (−1 + 3.73i)17-s + (−0.366 + 0.633i)19-s + (1.73 − 3.46i)20-s + (1 − i)22-s + (0.133 − 0.0358i)23-s + ⋯
L(s)  = 1  + (0.0947 + 0.353i)2-s + (0.750 − 0.433i)4-s + (0.834 − 0.550i)5-s + (−0.327 + 0.944i)7-s + (0.482 + 0.482i)8-s + (0.273 + 0.242i)10-s + (−0.411 − 0.713i)11-s + (0.554 − 0.554i)13-s + (−0.365 − 0.0262i)14-s + (0.308 − 0.533i)16-s + (−0.242 + 0.905i)17-s + (−0.0839 + 0.145i)19-s + (0.387 − 0.774i)20-s + (0.213 − 0.213i)22-s + (0.0279 − 0.00748i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.995 - 0.0932i$
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: \((2,\ 315,\ (\ :1/2),\ 0.995 - 0.0932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78246 + 0.0832579i\)
\(L(\frac12)\) \(\approx\) \(1.78246 + 0.0832579i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 1.23i)T \)
7 \( 1 + (0.866 - 2.5i)T \)
good2 \( 1 + (-0.133 - 0.5i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 2i)T - 13iT^{2} \)
17 \( 1 + (1 - 3.73i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.366 - 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.133 + 0.0358i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 4.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 + (-2.83 - 2.83i)T + 43iT^{2} \)
47 \( 1 + (8.83 - 2.36i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.83 + 6.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.09 + 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 + 2.86i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.26T + 71T^{2} \)
73 \( 1 + (12.9 + 3.46i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.83 + 1.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.09 - 2.09i)T - 83iT^{2} \)
89 \( 1 + (0.330 - 0.571i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.92 - 5.92i)T + 97iT^{2} \)
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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

−11.61331291831133812058860214280, −10.70659615657080396999755829247, −9.860791123889565491501277606386, −8.759196867736284944574600637088, −7.974913656203726401878413575969, −6.35771002083969034758203653722, −5.93675540895659970075214323457, −5.03112270028810569346207446293, −3.01665663749895562781597327469, −1.69014486281869312076594626809, 1.84696828558025605549915731406, 3.05564448238268555593243844399, 4.27738218916088437129536300274, 5.87407130476950780515077247836, 7.07562979378819235013823503953, 7.32713256083962367345237802799, 9.042880121619042558259303809947, 10.04819567101058126824130911335, 10.75053563311841962485077802165, 11.44719397816397154201708454783

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