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Properties

Label 1-304-304.195-r1-0-0
Degree $1$
Conductor $304$
Sign $0.995 - 0.0917i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
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  + (−0.342 + 0.939i)3-s + (0.984 − 0.173i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (0.866 − 0.5i)11-s + (−0.342 − 0.939i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.642 − 0.766i)21-s + (0.173 − 0.984i)23-s + (0.939 − 0.342i)25-s + (0.866 − 0.5i)27-s + (−0.642 + 0.766i)29-s + (0.5 − 0.866i)31-s + (0.173 + 0.984i)33-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)3-s + (0.984 − 0.173i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (0.866 − 0.5i)11-s + (−0.342 − 0.939i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)17-s + (−0.642 − 0.766i)21-s + (0.173 − 0.984i)23-s + (0.939 − 0.342i)25-s + (0.866 − 0.5i)27-s + (−0.642 + 0.766i)29-s + (0.5 − 0.866i)31-s + (0.173 + 0.984i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.995 - 0.0917i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.995 - 0.0917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.877272458 - 0.08634653040i\)
\(L(\frac12)\) \(\approx\) \(1.877272458 - 0.08634653040i\)
\(L(1)\) \(\approx\) \(1.128229219 + 0.1591252793i\)
\(L(1)\) \(\approx\) \(1.128229219 + 0.1591252793i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.984 - 0.173i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.20870187205966672922192578308, −24.184007303643284831518712653731, −23.3571524476913959419264320561, −22.53497638538516112225653048046, −21.685910472990706904378776278067, −20.559507252455063530729489228083, −19.43184274324971279064414643974, −18.942708693316818818534313907161, −17.568337012331627705552840433997, −17.20950524843484737514034941414, −16.404364275497118015720172084691, −14.66206841014867617672106099955, −13.940083518039198779309918800671, −13.19115884863820538893926115456, −12.25039017963238783570749220193, −11.24298117351320412893997162833, −10.06407146031114431843183045013, −9.29780483847618417382020447501, −7.80388308879880597161419835136, −6.76692172506424653308598023409, −6.28042618086718389374425904619, −4.972975544646507386686187469083, −3.489846593265794006194326555962, −1.98455816882465745649125651279, −1.14125715522493936694083115761, 0.668264181005801520965763597505, 2.50510959625679581673827651089, 3.50354154357297522849087506112, 5.02870841521912792229679988698, 5.716412334851276407361859139364, 6.56766274302918863665277183001, 8.40753072063085267076751426181, 9.36898365191395260227339978040, 9.88336768387549540893389259103, 11.005444202370688785367334332330, 12.09312729892091878473441336119, 12.98026702809539325276297427453, 14.328134240612582082966507395110, 14.93942237823039639655228547106, 16.18420894882575101393308835221, 16.733048775574390031181603105527, 17.71973661122609963898671994308, 18.61250856093737176292981467391, 19.86999107256605347973275311833, 20.813332262426656965011018058564, 21.593455558166162279915731619862, 22.342778190270783999794353502830, 22.83910976657326939804879262484, 24.490591131471786650342221940800, 25.10097283690580963601154089752

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