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Properties

Label 2-3460-3460.803-c0-0-0
Degree $2$
Conductor $3460$
Sign $0.873 + 0.487i$
Analytic cond. $1.72676$
Root an. cond. $1.31406$
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  + (0.889 − 0.457i)2-s + (0.581 − 0.813i)4-s + (0.0729 + 0.997i)5-s + (0.145 − 0.989i)8-s + (0.872 − 0.489i)9-s + (0.520 + 0.853i)10-s + (1.07 + 0.346i)13-s + (−0.322 − 0.946i)16-s + (−0.289 + 0.0318i)17-s + (0.551 − 0.833i)18-s + (0.853 + 0.520i)20-s + (−0.989 + 0.145i)25-s + (1.11 − 0.185i)26-s + (−1.20 + 1.34i)29-s + (−0.719 − 0.694i)32-s + ⋯
L(s)  = 1  + (0.889 − 0.457i)2-s + (0.581 − 0.813i)4-s + (0.0729 + 0.997i)5-s + (0.145 − 0.989i)8-s + (0.872 − 0.489i)9-s + (0.520 + 0.853i)10-s + (1.07 + 0.346i)13-s + (−0.322 − 0.946i)16-s + (−0.289 + 0.0318i)17-s + (0.551 − 0.833i)18-s + (0.853 + 0.520i)20-s + (−0.989 + 0.145i)25-s + (1.11 − 0.185i)26-s + (−1.20 + 1.34i)29-s + (−0.719 − 0.694i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3460\)    =    \(2^{2} \cdot 5 \cdot 173\)
Sign: $0.873 + 0.487i$
Analytic conductor: \(1.72676\)
Root analytic conductor: \(1.31406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3460} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3460,\ (\ :0),\ 0.873 + 0.487i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.428300890\)
\(L(\frac12)\) \(\approx\) \(2.428300890\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.889 + 0.457i)T \)
5 \( 1 + (-0.0729 - 0.997i)T \)
173 \( 1 + (0.946 - 0.322i)T \)
good3 \( 1 + (-0.872 + 0.489i)T^{2} \)
7 \( 1 + (-0.639 - 0.768i)T^{2} \)
11 \( 1 + (0.145 - 0.989i)T^{2} \)
13 \( 1 + (-1.07 - 0.346i)T + (0.813 + 0.581i)T^{2} \)
17 \( 1 + (0.289 - 0.0318i)T + (0.976 - 0.217i)T^{2} \)
19 \( 1 + (-0.0729 + 0.997i)T^{2} \)
23 \( 1 + (-0.145 - 0.989i)T^{2} \)
29 \( 1 + (1.20 - 1.34i)T + (-0.109 - 0.994i)T^{2} \)
31 \( 1 + (-0.872 - 0.489i)T^{2} \)
37 \( 1 + (-0.186 + 0.173i)T + (0.0729 - 0.997i)T^{2} \)
41 \( 1 + (-0.846 + 1.80i)T + (-0.639 - 0.768i)T^{2} \)
43 \( 1 + (0.946 + 0.322i)T^{2} \)
47 \( 1 + (-0.889 - 0.457i)T^{2} \)
53 \( 1 + (0.432 - 1.94i)T + (-0.905 - 0.424i)T^{2} \)
59 \( 1 + (-0.983 - 0.181i)T^{2} \)
61 \( 1 + (1.10 + 1.37i)T + (-0.217 + 0.976i)T^{2} \)
67 \( 1 + (0.489 + 0.872i)T^{2} \)
71 \( 1 + (0.994 - 0.109i)T^{2} \)
73 \( 1 + (-0.901 + 1.67i)T + (-0.551 - 0.833i)T^{2} \)
79 \( 1 + (0.889 - 0.457i)T^{2} \)
83 \( 1 + (-0.967 - 0.252i)T^{2} \)
89 \( 1 + (0.448 + 0.464i)T + (-0.0365 + 0.999i)T^{2} \)
97 \( 1 + (0.939 - 1.21i)T + (-0.252 - 0.967i)T^{2} \)
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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

−9.045458300342651194136330214140, −7.56228155535079104998745022276, −7.12351961786800470273970163551, −6.28152210570621462127414823501, −5.86270844651324638655387203174, −4.72508544174992499377461983990, −3.82645606262958238665416545874, −3.42174533582177802933244871267, −2.28366036157336527627264729762, −1.37780774059708800909612548537, 1.39710600532218776678065587528, 2.40340198834907977104699820674, 3.70839264917927297540422808946, 4.26635631130472675485558240954, 5.00638231203880030764303860530, 5.76635573947259800529653462258, 6.39708739342441338291455790489, 7.37262190723143291992383889621, 8.017848554388758124057013054252, 8.545812189917168630972362762551

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